Nonlinear drift wave evolution in the presence of a shear flow

Abstract:
Low frequency drift oscillations play an important role in the transport processes in magnetized plasmas, so they are intensely studied in recent decades [1]. The main problem in the drift waves' investigations is the presence of nonlinear effects even at relatively small amplitudes of waves. Moreover, vortex nonlinearity which is characteristic for the drift waves, is a degenerate one and their dispersion doesn't support the short wave components' moving away. Nonlinear generation of the high space
harmonics and their accumulation in the initial disturbance zone extremely complicate numerical simulations of the drift waves' evolution [2]. Thus some analytical approach based on symmetry analysis of the model is needed. In the present work, multiple-time-scale formalism of the perturbation theory is applied to the study of the nonlinear evolution of two drift waves, one of which is homogeneous in a drift direction and so represents a shear flow, and the second one is a standing wave which due to the degenerate nature of the vortex nonlinearity is an exact solution to the nonlinear equations of the model. So the temporal evolution of the waves in consideration is completely determined by their nonlinear interaction. Perturbation theory solution is obtained up to the third order in the waves' amplitude. Dependence of different harmonics' amplitudes and of the main harmonic frequency shift on the wave vector components of initial waves is determined. The results obtained show that nonlinear interaction of the considered system of waves produce a simultaneous generation of a large amount of high space harmonics
which amplitudes have strong anisotropic dependence on the components of the wave vectors of the initial state. The frequency shift of the main space harmonic is also anisotropic. Complicated calculations of the perturbation theory were controlled by the Mathematica package [3].

1. W. Horton, Drift waves and transport, Rev. Mod. Phys., vol. 3, pp.735-778, 1999.
2. V.Ya.Goloborod'ko, V.B.Taranov, Long term evolution of drift vortex structures,
   Journal of Plasma and Fusion Research, SERIES vol. 2, pp.335-338, 1999.
3. S.Wolfram, The Mathematica Book, version 3, Cambridge Univ. Press, 1996.